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• #### Appendix E: Sampling Variability and Uncertainty Analyses 163-172

LOGNORMAL CONFIDENCE LIMITS FOR THE MEAN

The Exact Method

Land (1971) developed an exact method for computing confidence limits for linear functions of the normal mean and variance. The classic example is the normalization of a lognormally distributed random variable x through the transformation , where, as noted previously, y is distributed normal with mean μ and variance σ2, or . Using Land’s (1975) tabled coefficients Hα, the one-sided (1 -α)100 percent lognormal LCL for the mean is

Alternatively, using H1-α, the one-sided (1 -α)100 percent lognormal UCL for the mean is

The factors H are given by Land (1975) and and sy are the mean and standard deviation of the natural log transformed data (i.e., ). Gilbert (1987) has a small subset of these extensive tables for n = 3 through 101, sy = .1 through 10.0, and α = .05 and .10 (i.e., upper and lower 90 percent and 95 percent confidence limit factors). Because these tables had historically been difficult to find, Gibbons and Coleman (2001) reproduced the complete set of Land’s (1975) tables and have also included computing approximations that can be used for automated applications. Land (1975) suggests that cubic interpolation (i.e., four-point Lagrangian interpolation) be used when working with these tables (Abramawitz and Stegun, 1964). A much easier and quite reasonable alternative is to use logarithmic interpolation.

Approximate Lognormal Confidence Limit Methods

There are also several approximations to lognormal confidence limits for the mean that have been proposed. These have been conveniently classified as either transformation methods or direct methods (Land, 1970). A transformation method is one in which the confidence limit is obtained for the expected value of some function of x and then transformed by some appropriate function to give an approximate limit for the

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